Coupled cluster

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Coupled cluster (CC) is a numerical technique used for describing many-body systems. Its most common use is as one of several quantum chemical post-Hartree-Fock ab initio quantum chemistry methods in the field of computational chemistry. It starts from the Hartree-Fock molecular orbital method and adds a correction term to take into account electron correlation. Some of the most accurate calculations for small to medium sized molecules use this method.

The method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear physics phenomena, but became more frequently used after Jiři Čížek and Josef Paldus reformulated the method for electron correlation in atoms and molecules in the 1960s. It is now one of the most prevalent methods in quantum chemistry that includes electronic correlation.

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[edit] Wavefunction ansatz

Coupled-cluster theory provides an approximate solution to the time-independent Schrödinger equation

\hat{H} \vert{\Psi}\rangle = E \vert{\Psi}\rangle

where \hat{H} is the Hamiltonian of the system. The wavefunction and the energy of the lowest-energy state are denoted by \vert{\Psi}\rangle and E, respectively. Other variants of the coupled-cluster theory, such as equation-of-motion coupled cluster and multi-reference coupled cluster may also produce approximate solutions for the excited states of the system.

The wavefunction of the coupled-cluster theory is written as an exponential ansatz:

\vert{\Psi}\rangle = e^{\hat{T}} \vert{\Phi_0}\rangle ,

where \vert{\Phi_0}\rangle is a Slater determinant usually constructed from Hartree-Fock molecular orbitals. \hat{T} is an excitation operator which, when acting on \vert{\Phi_0}\rangle, produces a linear combination of excited Slater determinants (see section below for greater detail).

The choice of the exponential ansatz is opportune because (unlike other ansatzes, for example, configuration interaction) it guarantees the size extensivity of the solution. Size consistency in CC theory, however, depends on the size consistency of the reference wave function.

[edit] Cluster operator

The cluster operator is written in the form

\hat{T}=\hat{T}_1 + \hat{T}_2 + \hat{T}_3 + \cdots ,

where \hat{T}_1 is the operator of all single excitations, \hat{T}_2 is the operator of all double excitations and so forth. In the formalism of second quantization these excitation operators are conveniently expressed as

\hat{T}_1=\sum_{i}\sum_{a} t_{i}^{a} \hat{a}_{i}\hat{a}^{\dagger}_{a},
\hat{T}_2=\frac{1}{4}\sum_{i,j}\sum_{a,b} t_{ij}^{ab} \hat{a}_{i}\hat{a}_j\hat{a}^{\dagger}_{a}\hat{a}^{\dagger}_{b}

etc

In the above formulae \hat{a}^{\dagger} and \hat{a} denote the creation and annihilation operators respectively and i, j stand for occupied and a, b for unoccupied orbitals. The creation and annihilation operators in the coupled cluster terms above are written in canonical form, where each term is in normal order. Being the one-particle excitation operator and the two-particle excitation operator, \hat{T}_1 and \hat{T}_2 convert the reference function \vert{\Phi_0}\rangle into a linear combination of the singly- and doubly-excited Slater determinants, respectively. Solving for the unknown coefficients t_{i}^{a} and t_{ij}^{ab} is necessary for finding the approximate solution \vert{\Psi}\rangle.

Taking into consideration the structure of \hat{T}, the exponential operator e^{\hat{T}} may be expanded into Taylor series:

e^{\hat{T}} = 1 + \hat{T} + \frac{\hat{T}^2}{2!} + \cdots = 1 + \hat{T}_1 + \hat{T}_2 + \frac{\hat{T}_1^2}{2} + \hat{T}_1\hat{T}_2 + \frac{\hat{T}_2^2}{2} + \cdots

This series is finite in practice because the number of occupied molecular orbitals is finite, as is the number of excitations. In order to simplify the task for finding the coefficients t, the expansion of \hat{T} into individual excitation operators is terminated at the second or slightly higher level of excitation (rarely exceeding four). This approach is warranted by the fact that even if the system admits more than four excitations, the contribution of \hat{T}_5, \hat{T}_6 etc to the operator \hat{T} is small. Furthermore, if the highest excitation level in the \hat{T} operator is n,

T = 1 + \hat{T}_1 + ... + \hat{T}_n

then Slater determinants excited more than n times may (and usually do) still contribute to the wave function \vert{\Psi}\rangle because of the non-linear nature of the exponential ansatz. Therefore, coupled cluster terminated at \hat{T}_n usually recovers more correlation energy than configuration interaction with maximum n excitations.

[edit] Coupled-cluster equations

Coupled-cluster equations are equations whose solution is the set of coefficients t. There are several ways of writing such equations but the standard formalism results in a terminating set of equations which may be solved iteratively. The naive variational approach does not take advantage of the connected nature of the cluster amplitudes and results in a non-terminating set of equations.

Suppose there are q coefficients t to solve for. Therefore, we need q equations. It is easy to notice that each t-coefficient may be put in correspondence with a certain excited determinant: t_{ijk...}^{abc...} corresponds to the determinant obtained from \vert{\Phi_0}\rangle by substituting the occupied orbitals i,j,k,... with the virtual orbitals a,b,c,... Projecting the Schrödinger equation with the exponential ansatz by q such different determinants from the left, we obtain the sought-for q equations:

\langle {\Psi^{*}}\vert \hat{H} e^{\hat{T}} \vert{\Psi_0}\rangle = E \langle {\Psi^{*}} \vert e^{\hat{T}} \vert {\Psi_0}\rangle

where by \vert{\Psi^{*}}\rangle we understand the whole set of the appropriate excited determinants.

Unfortunately, \langle {\Psi^{*}} \vert e^{\hat{T}} \vert {\Psi_0}\rangle is a non-terminating sequence. The coupled-cluster equations are reduced to a closed form in the similarity transformed representation:

E_{CORR} = \langle {\Psi_0}\vert e^{-\hat{T}} \hat{H}_N e^{\hat{T}} \vert{\Psi_0}\rangle = \langle {\Psi_0}\vert \bar{H}_N \vert{\Psi_0}\rangle,
0 = \langle {\Psi^{*}}\vert e^{-\hat{T}} \hat{H}_N e^{\hat{T}} \vert{\Psi_0}\rangle ,

the latter being the equations to be solved and the former the equation for the evaluation of the energy. Consider the standard CCSD method:

0 = \langle {\Psi_0}\vert e^{-(\hat{T}_1+\hat{T}_2)} \hat{H}_N e^{(\hat{T}_1+\hat{T}_2)} \vert{\Psi_0}\rangle ,
0 = \langle {\Psi_{S}}\vert e^{-(\hat{T}_1+\hat{T}_2)} \hat{H}_N e^{(\hat{T}_1+\hat{T}_2)} \vert{\Psi_0}\rangle ,
0 = \langle {\Psi_{D}}\vert e^{-(\hat{T}_1+\hat{T}_2)} \hat{H}_N e^{(\hat{T}_1+\hat{T}_2)} \vert{\Psi_0}\rangle ,

which when reduced using the BCH formula for the similarity transformed Hamiltonian, e^{\hat{-T}} \hat{H} e^{\hat{T}} = H + [H,T] + (1/2)[[H,T],T] + ... , become quartic in T1 and quadratic in T2. Standard quantum chemistry packages (ACES, NWChem, etc.) solve the coupled-equations using as Ax = 0 , where A is the coupled-cluster Jacobian and x is the vector of amplitudes. In general, the resulting similarity transformed Hamiltonian may not be hermitian.

[edit] Types of coupled-cluster methods

The classification of traditional coupled-cluster methods rests on the highest number of excitations allowed in the definition of \hat{T}. The abbreviations for coupled-cluster methods usually begin with the letters "CC" (for coupled cluster) followed by

  1. S - for single excitations (shortened to singles in coupled-cluster terminology)
  2. D - for double excitations (doubles)
  3. T - for triple excitations (triples)
  4. Q - for quadruple excitations (quadruples)

Thus, the \hat{T} operator in CCSDT has the form

T = \hat{T}_1 + \hat{T}_2 + \hat{T}_3.

Terms in round brackets indicate that these terms are calculated based on perturbation theory. For example, a CCSD(T) approach simply means:

  1. A coupled-cluster method
  2. It includes singles and doubles fully
  3. Triples are calculated with perturbation theory.

[edit] General description of the theory

The complexity of equations and the corresponding computer codes, as well as the cost of the computation increases sharply with the highest level of excitation. For many applications the sufficient accuracy may be obtained with CCSD, and the more accurate (and more expensive) CCSD(T) is often called "the gold standard of quantum chemistry" for its excellent compromise between the accuracy and the cost for the molecules near equilibrium geometries. More complicated coupled-cluster methods such as CCSDT and CCSDTQ are used only for high-accuracy calculations of small molecules. The inclusion of all n levels of excitation for the n-electron system gives the exact solution of the Schrödinger equation within the given basis set.

One possible improvement to the standard coupled-cluster approach is to add terms linear in the interelectronic distances through methods such as CCSD-R12. This improves the treatment of dynamical electron correlation by satisfying the Kato cusp condition and accelerates convergence with respect to the orbital basis set. Unfortunately, R12 methods invoke the resolution of the identity which requires a relatively large basis set in order to be a good approximation.

The coupled-cluster method described above is also known as the single-reference (SR) coupled-cluster method because the exponential ansatz involves only one reference function \vert{\Phi_0}\rangle. The standard generalizations of the SR-CC method are the multi-reference (MR) approaches: state-universal coupled cluster (also known as Hilbert space coupled cluster), valence-universal coupled cluster (or Fock space coupled cluster) and state-selective coupled cluster (or state-specific coupled cluster).

[edit] A historical account

In the first reference below, Kümmel comments:

Considering the fact that the CC method was well understood around the late fifties it looks strange that nothing happened with it until 1966, as Jiři Čížek published his first paper on a quantum chemistry problem. He had looked into the 1957 and 1960 papers published in Nuclear Physics by Fritz and myself. I always found it quite remarkable that a quantum chemist would open an issue of a nuclear physics journal. I myself at the time had almost gave up the CC method as not tractable and, of course, I never looked into the quantum chemistry journals. The result was that I learnt about Jiři's work as late as in the early seventies, when he sent me a big parcel with reprints of the many papers he and Joe Paldus had written until then.

[edit] Relation to other theories

[edit] References

  • H.G. Kümmel, A biography of the coupled cluster method - found in R.F. Bishop, T. Brandes, K.A. Gernoth, N.R. Walet, Y. Xian (Eds.), Recent progress in many-body theories, Proceedings of the 11th international conference, World Scientific Publishing, Singapore, 2002, pp. 334-348.
  • Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd., 191 - 232. ISBN 0-471-48552-7. 

[edit] External resources

  • Cramer, C. J. Essentials of Computational Chemistry: Theories and Models (Wiley and Sons)